Let A be a strict coherent analyitc geometires. Proposition 5.5.1. (a) Suppose W
X is a proper strong subobject. There is a prime V of X
such that the localization X_{V} is not contained in W.
Proof. Since any proper strong subobject is contained in a proper
regular subobject, and any proper regular subobject is an intersection
of proper finitely cogenerated regular subobjects by (4.1.3),
it suffices to prove the assertion for a proper finitely cogenerated regular
subobject W. First we note that there is a prime V of X
such that any analytic neighborhood U of V is not contained
in W. Otherwise W contains an analytic cover {U_{i}}
of X, which is not the case as the category is strict, therefore
X is the colimit of {U_{i}
U_{j}}, and this only happens if X = W. Thus let V
be such a prime. The localization X_{V} of X at V
is the cofiltered limit of all the analytic neighborhoods of V .
Assume X_{V} is contained in W. Suppose W
is the equalizer of a pair (r_{1}, r_{2}): X
> T of maps with T finitely copresentable. There is an open
neighborhood u: U > X of V such that r_{1}v
= r_{2}v (see the proof of (4.1.4)).
Then V
U which contradicts to the choice of V. This shows
that X_{V} is not contained in W.
Proposition 5.5.2. An object is reduced iff each of its localization is reduced. Proof. Suppose X is reduced. By (3.1.3.e)
analytic subobjects of X are reduced. Any localization of X
is a cofiltered limit of analytic subobjects of X, thus is reduced
by (5.1.5.b).
Proposition 5.5.3. A map f: Y > X is epic iff its pullback along any localization of X is so. Proof. The pullback of any epi along a localization is epic as
any localization is coflat.
Proposition 5.5.4. Suppose f:
Y > X is a mono.
Proof. (a) Suppose P
Spec(Y) and Q
Spec(Y) are two primes over the same O
Spec(X). Then their residues k(P) and
k(Q) are isomorphic to the residue k(O) of
O by assumption. Thus there are isomorphisms u: k(P)
> k(O) and v: k(Q) > k(O).
Let s (resp. t) be the compositions of u^{1}:
k(O) > k(P) (resp. v^{1}:
k(O) > k(Q)) with the inclusions k(P)
> Y (resp. k(Q) > Y ). Then fs = ft is
the inclusion k(O) > X. Since f is a mono,
we have s = t. So P = Q. This shows that Spec(f)
is injective.
Proposition 5.5.5. Suppose f:
Y > X is a map.
Proof. (a) If Spec(f) is not surjective we can
find a residue P > X which does not factors through f,
then P > X is disjoint with f, thus f is not unipotent.
Proposition 5.5.6. Any unipotent local isomorphic mono f: Y > X is an isomorphism. Proof. First note that Spec(f) is bijective by (5.5.4) and (5.5.5). Let Z be the coproducts of all localization l_{P}: Y_{P} > Y and let z: Z > Y be the map induced by l_{P}, then z is an epi by (5.5.1.b). Since f is a local isomorphism and Spec(f) is bijective, Z is also naturally the coproducts of localizations of X with fz: Z > X as the canonical map. Thus fz is epic by (5.5.1.b), which implies that f is epic. Since finitely copresentable objects form a strong generating set of A, to see that f is an isomorphism, it suffices to prove that any map t: Y > C from Y to a finitely copresentable object C factors through f. Suppose p: Y_{P} > Y is a localization of Y at a prime P. Since f is a local isomorphism, fp: Y_{P} > X is the localization of X at f^{+1}(P). Thus Y_{P} is the cofiltered limit of a collection of analytic neighborhoods V_{i} of f^{+1}(P) with the maps (fp)_{s}: Y_{P} > V_{i}. Since C is finitely copresentable, there exists an analytic neighborhood V_{s} of f^{+1}(P) and a map g_{s}: V_{s} > C such that t_{s} p_{s} = g_{s} f_{s}p_{s}, where t_{s}: f^{1}(V_{s}) > C, p_{s}: Y_{P} > f^{1}(V_{s}) and f_{s}: f^{1}(V_{s}) > V_{s} are the induced maps. Since C is finitely copresentable and Y_{P} is the filtered limits of open analytic neighborhood of P, and f^{1}(V_{s}) is one of such analytic neighborhood, there is a small analytic neighborhood U_{P} contained in f^{1}(V_{s}) such that the restrictions of t_{s} and g_{s} f_{s} on U_{P} are the same, denoted this restriction by t_{P} . We have proved that for any prime P we can find a small open neighborhood U_{P} of P such that the restriction of t on U_{P} can be factored through the restriction of f on U_{P}. Since the analytic category is strict, these factorization can be glue together to obtain a global factorization for t by f. Proposition 5.5.7. A mono is a local isomorphism iff it is a fraction. Proof. Suppose u: U > X is a local isomorphic
mono. Any unipotent pullback of u is a unipotent local isomorphic,
therefore is an isomorphism by (5.5.6). Thus u
is normal. So we only need to prove that u is coflat. Since the
class of local isomorphisms is closed under pullback, it suffices to prove
that u is precoflat. Suppose t: T > X is an epi
and r: Z > U, s: Z > T is the pullback
of f and t. For any localization g: G > U
the map r^{1}(U) > U is epic because it
is the pullback of the epic map t along the localization ug:
G > X. This shows that if r factors through a strong mono
w: W U, then
W contains all the localization of U. This is only possible
if W = U by (5.5.1.a). This shows that r
is epic, which means that u is precoflat as desired. The other
direction has been proved in (5.3.3).
